"wings" appear when I add a subsurface modifier to some simple geometry
Ngons (faces with more than 4 sides)

Join some of those vertices (manually subdivide edges where needed) so that you have at most 4 sides for each face.
You have a lot going on in this scene. Blender can highlight ngons for you.
In Edit mode, go to Select -&gt; Select All By Trait -&gt; Faces By Sides.
A little menu will appear in the lower-left. Choose 4 and Greater Than.
Do this with the Subdivision Surface modifier turned off, it will be easier.
Here is a .gif showing you how to isolate islands to work on (subdivision surface modifier has been turned off)